marcel tried to use cavalieris principle to show that the two figures have the same volume. he base areas are the same. therefore, corresponding cross - sections have the same area. so the volumes must be the same.\ what is the first mistake marcel made? choose 1 answer: a the base areas are not the same. b it isnt true that corresponding cross - sections have the same area. c he did not establish that the heights are the same.

Question

marcel tried to use cavalieris principle to show that the two figures have the same volume. 	he base areas are the same. therefore, corresponding cross - sections have the same area. so the volumes must be the same.\ what is the first mistake marcel made? choose 1 answer: a the base areas are not the same. b it isnt true that corresponding cross - sections have the same area. c he did not establish that the heights are the same.

Accepted Answer

# Brief Explanations:
Cavalieri's principle states that if two solids have the same height and the same cross - sectional area at every level (corresponding cross - sections), then they have the same volume. Marcel only considered the base area and assumed corresponding cross - sections have the same area. But he forgot a key part of the principle: the heights of the two figures must also be the same. Option A is wrong because the base areas (both are circles with radius 4, so $A=\pi r^{2}=\pi	imes4^{2}$) are the same. Option B is wrong because if the base areas are the same and the figures are such that the cross - sectional area depends only on the height (like in a cylinder, the cross - section at any height is a circle with the same radius as the base), but in this case, the second figure is a frustum - like or a different shape. Wait, no, actually, for the two figures (a cylinder and a frustum or a cone - frustum), the cross - sectional area at different heights: in the cylinder, the cross - section is always a circle with radius 4. In the other figure, as we move up, the radius of the cross - section may change (since it's a wider - at - the - top shape). Wait, no, the main mistake is about the height. Wait, no, let's re - evaluate. The two figures: one is a cylinder (vertical sides), one is a shape with slanted sides (like a frustum of a cone or a different prismatoid). The base area: both have a circular base with radius 4, so base area is same ($\pi r^{2}$). But for Cavalieri's principle, we need that at every height (from the base up to the top), the cross - sectional area is the same. But in the non - cylinder figure, as we move up, the radius of the cross - section (if it's a circular cross - section) would be larger than 4 (since the sides are slanting out). Wait, no, the figure on the right has a wider top. So the cross - sectional area at a height h from the base: in the cylinder, it's $\pi	imes4^{2}$. In the right - hand figure, at height h, the radius of the cross - section is larger than 4 (since the sides are slanting out), so the cross - sectional area is larger. So Marcel's mistake is in assuming that corresponding cross - sections have the same area. Wait, but the options: Option B says "It isn't true that corresponding cross - sections have the same area." But wait, the base area is same (both circles with r = 4). Wait, maybe I misread the figures. Wait, the first figure is a cylinder (vertical sides), the second is a shape with the base radius 4 and the top radius larger (since it's flared out). So at the base (height 0), cross - section area is same. But at a height above the base, in the cylinder, cross - section is still radius 4, in the second figure, cross - section radius is more than 4, so cross - section area is larger. So Marcel's first mistake is thinking that corresponding cross - sections have the same area. But wait, the options: Wait, no, let's check the options again. Option C: "He did not establish that the heights are the same." Wait, maybe the two figures have different heights? But the diagram shows the dashed line (height) as the same? Wait, the problem is about the application of Cavalieri's principle. The correct answer is B? No, wait, let's recall the principle. The principle requires that for every plane parallel to the base, the cross - sectional area is equal. In the cylinder, all cross - sections are circles with radius 4. In the other figure (the one with slanted sides), the cross - sections at different heights will have different radii (larger than 4 as we go up), so their areas will be larger than $\pi	imes4^{2}$. So Marcel's mistake is in saying that corresponding cross - sections have the same area. But wait, the options: Option B says that. But wait, the base areas are same (both $\pi r^{2}$ with r = 4). So Option A is wrong. Option C: if the heights are the same, but the cross - sections are different, then the principle doesn't apply. Wait, maybe I made a mistake. Let's re - read the options. Option B: "It isn't true that corresponding cross - sections have the same area." This is correct. Because in the cylinder, cross - section is always radius 4. In the other figure, as we move up, the radius of the cross - section increases (since the sides are slanting out), so the cross - sectional area is larger. So Marcel assumed that just because base areas are same, corresponding cross - sections are same, which is wrong.
# Answer:
B. It isn't true that corresponding cross - sections have the same area.

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